A rambling narrative of one man's journey through mathematics

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Last week was the first of the big 3-manifolds events at IAS and overall, it was spectacular. The highlight, without a doubt, was Dave Gabai being amazing during the last talk of the week, but there were some other great moments too…

…and some not-so-great ones, including some woman whom I don’t know interrupting Genevieve Walsh‘s talk no fewer than 10 times to say random rude things about how it was not-good (which was untrue), unoriginal (only true in the sense that Dr. Walsh spent some time talking about general background that she didn’t claim to have invented), and a waste of time. I was pretty blown away that such things happened at pure math talks, but I guess pure math people are people too and – at the end of the day – people just look for a way to disappoint and/or bring down other people.

I learned a lot, though, and I came away with a new direction for my own research, so that’s going to be the goal moving forward: To balance the somewhat-regular yearly 3-manifolds talks at IAS with the stuff I need to figure out to get my own stuff knocked out.

Oh, and plus side: I actually got a full week of salaried work done! YAY FOOD! But the downside is that I’m having to drop $2k on random car things (making our tires able to withstand rain and snow and making it so that our heat keeps hypothermia at bay), so…YAY CREDIT CARD DEBT! ::wink::

Alright, well I’m awake for some dumb reason so I guess I’ll…try to do something…constructive. Or something. Hah.

My previous topology classes were taught by someone who’s amazing <i>as a mathematician</i>. Most people in the class would agree, however, that this person was someone who was terrible as an instructor: Somehow, I made it through a graduate sequence of coursework despite receiving terrible grades throughout. This happened despite my spending 50+ hours a week on every homework assignment and slaving until I was on the verge of breakdown week in and week out. Somehow, I got a B.

My topology experience thus far is certainly not one of my finest achievements.

Fast forward to now and I’m in an algebraic topology class taught by someone who’s amazing. Amazing. Not amazing <i>insert quantifier here</i>, no – this person is simply amazing. And this topic is beautiful. And this class is hard.

This class is hard, too, despite the fact that we have no continual responsibilities. Indeed, we have zero homework whatsoever: Not required problems to turn in, not required problems to keep, not even suggested problems for our benefit. We simply have <i>zero homework</i> in this class. That’s a huge relief after last semester.

What we <i>do have</i>, though, are exams. We have three of them, and I have zero doubt now (nor have I had doubt at any point this semester) that my ass will be kicked by each and every one.

As a result, I’m working hard.

A week-ish ago, I spent some time going through the preliminary parts of the stuff we’re talking about (homology theory). I did examples, I spent lots of time drawing pictures, and I didn’t stop until I got it.

That’s right: A week-ish ago, I <i>got it</i>.

Today, however, I’m sitting in my office, frustrated and almost-defeated, blogging to you all and mourning the fact that a lot has apparently changed in the last week-ish.

Today, I just don’t get it.

If I were to make a list here cataloging the number of screw-ups I made trying to solve one problem over the course of about 20 hours, I’d be (a) making a really long list and (b) really <i>really</i> embarrassed.

I’m really <i>really</i> embarrassed right now.

Finally, after re-reading and re-re-reading Hatcher, I found source 1 of my confusion. Later, after consulting the online resources of mathematicians greater than myself (case in point here), I found the remaining sources of my confusion.

The upside is that now I’m no longer confused. On the other hand, the fact that I was as confused as I was (and about such basic material as that happened to be) makes me really <i>really</i> uneasy moving forward.

I need an intervention.

In the meantime, I’m going to try to dust myself off, hit the salt mines yet again, and lose my frustrations in the never-ending cycle of Lana del Rey that’s been permeating through my office for the past couple hours.

I was in my fourth semester (a spring semester) as a master’s student: I had passed my two mandatory abstract algebra classes my first two semesters there and had passed my comprehensive exams during the Fall (my third semester). As was custom, then, I spent my third and fourth semesters taking random “advanced topics” courses aimed at potential doctoral students, and one of the sequences I took was the algebra sequence.

My first semester doctoral-level (or 7000-level as was colloquial there) algebra class was over the classification of finite simple groups and was by far the most difficult class I’d ever taken at the time. Apparently, being a student who doesn’t remotely have the sufficient background and being in a class run by a professor who has unimaginably-greater background – who teaches as if the audience consists of peers – makes for a difficult time. I squeaked out an A.

In the second semester of 7000-algebra, however, things were far less directed. Long story short, it was a potpurri of material, some from algebraic topology, some from homological algebra, and some – about 1/3 of the course, I’d say – from category theory. That was my very first exposure to an area I didn’t otherwise know existed and I remember thinking, This is the most abstract thing that’s ever been devised, and also, There’s no way this will ever be far-reaching outside the realm of mathematics.

I’ve since realized that the first assertion isn’t really true – unsurprisingly since my exposure to other areas has increased drastically since leaving there – but apparently, the second one isn’t either. To be more precise, I stumbled upon this article online which describes a number of non-math areas that have been benefiting – and will continue to benefit – from the use of category theoretic ideas.

It’s really quite amazing to see, but in and of itself is unsurprising given the fact that category theory itself was devised to provide unity among the wide variety of subdisciplines of mathematics. As a pure mathematician, I always tried to find a balance between being interested in too broad a range of topics and being too narrow with my scope; the spread of category theory invites us all to analyze that aspect of ourselves. To borrow a quote from David Spivak’s exposition (available on the arXiv),:

It is often useful to focus ones study by viewing an individual thing, or a group of things, as though it exists in isolation. However, the ability to rigorously change our point of view, seeing our object of study in a diﬀerent context, often yields unexpected insights. Moreover this ability to change perspective is indispensable for eﬀectively communicating with and learning from others. It is the relationships between things, rather than the things in and by themselves, that are responsible for generating the rich variety of phenomena we observe in the physical, informational, and mathematical worlds.

So I was able – fortunately – to wake up early and to do some legit reading, despite having only a handful of sleep hours (4-ish?). That’s a definite positive. Right now, I’m about 30 minutes away from a forced obligation (that’s a definite negative), but I wanted to use the 30 minutes I have to still do something constructive. Rather than spend this time wracking my brain with really difficult, hard-to-understand reading that would leave me mentally exhausted for the aforementioned obligation, I decided to come here and write a little exposition regarding something mathematical.

In particular, I’m going to talk about the so-called Richard’s Paradox (see here).

Of course, the fact that I’m avoiding theoretical math to postpone mental exhaustion while using the time to come here and talk about theoretical math is a bit of a paradox as well, so I’ll basically be expositing, paradoxically, about paradoxes.

You have no idea how much I crack myself up.

The ideology that birthed Richard’s paradox is intimately tied to the idea of metamathematics, that is, the study of metatheories – theories about mathematical theories – using mathematical ideas and quantification. I’m not going to get too deeply involved in the discussion on that particular topic; the interested reader, of course, can scope out more here.

To begin, we let denote the set of nonzero positive integers (aka, the natural numbers) and we investigate the collection of all “formal English language statements of finite length” which define a number of . For example, The first prime number, The smallest perfect number, and The cube of the first odd number larger than five are such statements, as they verbally describe the numbers 2, 6, and 73=343, respectively. On the other hand, statements like The number larger than all other numbers and Scotland is a place I’d like to visit fail to make the list due to the fact that the first doesn’t describe a number in and the second doesn’t describe a number at all. Let denote the collection of all so-called qualifying statements, that is, statements that do describe elements .

Note, first, that the collection is infinite due to the fact that the statements The ith natural number is a qualifying statement for all . It’s also countable: Only a countable number of words exist in the English language, and each statement in consists of a finite union of these countably many words. This fact, along with obvious language considerations, says that can actually be given an ordering.

Indeed, consider a two-part ordering: First, organize the statements in by length so that the shortest statements appear first, and then organize statements of the same length by standard lexicographical (dictionary) ordering. The result is an ordered version of the countably infinite collection which we’ll again denote by .

"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos